Can the co-domain be any set containing the range?

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When you choose the co-domain can it just be anything that is greater than the range (i.e. the image)? In general, should the co-domain just be whatever space it maps to: R^2, R^3, etc. whereas the range is a subset of that space such as R^+ (all positive reals)?
 

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Yes, the codomain is the space that the function maps to - that is, all the possible outcomes. This is usually given in the definition of the function. The image/range is what actually comes out. The wikipedia page for http://en.wikipedia.org/wiki/Codomain" has some examples
 
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